Subgroups — C₃×#3155 #85019

Order 384.

596 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#535	C₂	1	2	yes	maximal, complemented
5	C₃×#496	C₂	1	4	yes	maximal, complemented
6	C₃×#495	C₂	1	2	yes	maximal, complemented
7	C₃×#493	C₂	1	2	yes	maximal, complemented
8	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	#3155	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	C₂²	1	2	yes	—
11	C₂²×C₃×D₄	C₂²	1	2	yes	—
12	C₃×AGL(1,ℤ/8)	—	2	8	no	—
13	C₃×#135	—	2	4	no	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
16	C₂×C₃×SD₁₆	C₂²	1	4	yes	complemented
17	C₂×C₃×D₈	—	2	4	no	—
18	C₂×C₃×D₈	C₂²	1	2	yes	—
19	C₂×C₃×D₈	C₂²	1	2	yes	—
20	C₂×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×D₈	C₂²	1	4	yes	complemented
22	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
24	C₃×#127	C₂²	1	4	yes	complemented
25	C₃×#124	C₂²	1	4	yes	complemented
26	C₃×#121	—	2	4	no	—
27	C₃×#120	—	2	4	no	—
28	C₃×#117	C₂²	1	1	yes	characteristic
29	C₃×#97	C₂²	1	2	yes	complemented
30	C₃×C₄×C₈	C₂²	1	2	yes	complemented
31	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
32	C₂×#135	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂²×D₈	C₂×C₃	1	1	yes	characteristic, complemented
34	#535	C₂×C₃	1	2	yes	complemented
35	#496	C₂×C₃	1	4	yes	complemented
36	#495	C₂×C₃	1	2	yes	complemented
37	#493	C₂×C₃	1	2	yes	complemented
38	#405	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	—	2	2	no	—
40	C₃×#41	—	2	4	no	—
41	C₃×#41	—	2	4	no	—
42	C₂×C₃×Q₈	C₂³	1	2	yes	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	4	no	—
45	C₂×C₃×D₄	—	2	4	no	—
46	C₂×C₃×D₄	—	2	2	no	—
47	C₂×C₃×D₄	—	2	8	no	—
48	C₂×C₃×D₄	C₂³	1	4	yes	—
49	C₂×C₃×D₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₃×Q₁₆	—	2	2	no	—
53	C₃×SD₁₆	—	2	4	no	—
54	C₃×SD₁₆	—	2	8	no	—
55	C₃×D₈	—	2	4	no	—
56	C₃×D₈	—	2	4	no	—
57	C₃×D₈	—	2	2	no	—
58	C₃×D₈	—	2	8	no	—
59	C₃×M₁₆	D₄	1	4	yes	complemented
60	C₂×C₃×C₈	D₄	1	4	yes	complemented
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	2	yes	—
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₃×C₄⋊C₄	D₄	1	2	yes	—
65	C₃×C₂²⋊C₄	—	2	8	no	—
66	C₃×C₂²⋊C₄	D₄	1	2	yes	—
67	C₃×C₄²	C₂³	1	2	yes	—
68	C₂×#41	C₂²×C₃	1	2	yes	—
69	C₂²×D₄	C₂²×C₃	1	2	yes	—
70	AGL(1,ℤ/8)	—	2	8	no	—
71	#135	—	2	4	no	—
72	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
73	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
74	C₂×SD₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×D₈	—	2	4	no	—
76	C₂×D₈	C₂²×C₃	1	2	yes	—
77	C₂×D₈	C₂²×C₃	1	2	yes	—
78	C₂×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₂×D₈	C₂²×C₃	1	4	yes	complemented
80	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
82	#127	C₂²×C₃	1	4	yes	complemented
83	#124	C₂²×C₃	1	4	yes	complemented
84	#121	—	2	4	no	—
85	#120	—	2	4	no	—
86	#117	C₂²×C₃	1	1	yes	characteristic
87	#97	C₂²×C₃	1	2	yes	complemented
88	C₄×C₈	C₂²×C₃	1	2	yes	complemented
89	C₂³×C₃	—	2	4	no	—
90	C₂³×C₃	—	2	2	no	—
91	C₂³×C₃	—	4	4	no	—
92	C₂³×C₃	—	8	4	no	—
93	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
94	C₃×Q₈	—	2	2	no	—
95	C₃×Q₈	—	2	2	no	—
96	C₃×D₄	—	2	4	no	—
97	C₃×D₄	—	2	2	no	—
98	C₃×D₄	—	2	4	no	—
99	C₃×D₄	—	2	2	no	—
100	C₃×D₄	—	4	4	no	—
101	C₃×D₄	—	4	4	no	—
102	C₃×D₄	—	8	8	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	4	4	no	—
105	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
110	C₃×C₈	C₂×D₄	1	2	yes	—
111	C₃×C₈	C₂×D₄	1	2	yes	—
112	C₃×C₈	C₂×D₄	1	4	yes	—
113	C₂⁴	—	2	2	no	—
114	#41	—	2	4	no	—
115	#41	—	2	4	no	—
116	C₂×Q₈	C₂³×C₃	1	2	yes	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	—	2	4	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	8	no	—
122	C₂×D₄	C₂³×C₃	1	4	yes	—
123	C₂×D₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	Q₁₆	—	2	2	no	—
127	SD₁₆	—	2	4	no	—
128	SD₁₆	—	2	8	no	—
129	D₈	—	2	4	no	—
130	D₈	—	2	4	no	—
131	D₈	—	2	2	no	—
132	D₈	—	2	8	no	—
133	M₁₆	C₃×D₄	1	4	yes	complemented
134	C₂×C₈	C₃×D₄	1	4	yes	complemented
135	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₈	C₂³×C₃	1	2	yes	—
137	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
138	C₄⋊C₄	C₃×D₄	1	2	yes	—
139	C₂²⋊C₄	—	2	8	no	—
140	C₂²⋊C₄	C₃×D₄	1	2	yes	—
141	C₄²	C₂³×C₃	1	2	yes	—
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	—	4	2	no	—
144	C₂²×C₃	—	4	4	no	—
145	C₂²×C₃	—	4	4	no	—
146	C₂²×C₃	—	4	8	no	—
147	C₂²×C₃	—	8	8	no	—
148	C₂²×C₃	#127	1	2	yes	—
149	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
150	C₃×C₄	—	2	4	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	#127	1	2	yes	—
153	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	4	4	no	—
158	C₂³	—	8	4	no	—
159	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
160	Q₈	—	2	2	no	—
161	Q₈	—	2	2	no	—
162	D₄	—	2	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	4	no	—
165	D₄	—	2	2	no	—
166	D₄	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	8	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
176	C₈	C₂×C₃×D₄	1	2	yes	—
177	C₈	C₂×C₃×D₄	1	2	yes	—
178	C₈	C₂×C₃×D₄	1	4	yes	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	—	8	2	no	—
181	C₂×C₃	—	8	4	no	—
182	C₂×C₃	C₂×#127	1	1	yes	characteristic
183	C₂×C₃	#576	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	—	4	8	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	C₃×#127	1	2	yes	—
191	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	4	2	no	—
194	C₄	C₃×#127	1	2	yes	—
195	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
196	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₃	#3155	1	1	yes	characteristic, direct product factor, Sylow
198	C₂	—	2	2	no	—
199	C₂	—	8	4	no	—
200	C₂	—	8	2	no	—
201	C₂	C₂×C₃×#127	1	1	yes	characteristic
202	C₂	C₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₃×#135	C₂	semidirect
3. C₂²×C₃×D₈	C₂	semidirect
4. C₃×#535	C₂	semidirect
5. C₃×#496	C₂	semidirect
6. C₃×#495	C₂	semidirect
7. C₃×#493	C₂	semidirect
8. C₃×#405	C₂	semidirect
9. #3155	C₃	direct (N × H)
14. C₂×C₃×Q₁₆	C₂²	semidirect
15. C₂×C₃×SD₁₆	C₂²	semidirect
16. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₂×C₃×D₈	C₂²	semidirect
21. C₂×C₃×D₈	C₂²	semidirect
24. C₃×#127	C₂²	semidirect
25. C₃×#124	C₂²	semidirect
29. C₃×#97	C₂²	semidirect
30. C₃×C₄×C₈	C₂²	semidirect
31. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
32. C₂×#135	C₂×C₃	semidirect
33. C₂²×D₈	C₂×C₃	semidirect
34. #535	C₂×C₃	semidirect
35. #496	C₂×C₃	semidirect
36. #495	C₂×C₃	semidirect
37. #493	C₂×C₃	semidirect
38. #405	C₂×C₃	semidirect
59. C₃×M₁₆	D₄	semidirect
60. C₂×C₃×C₈	D₄	semidirect
72. C₂×Q₁₆	C₂²×C₃	semidirect
73. C₂×SD₁₆	C₂²×C₃	semidirect
74. C₂×SD₁₆	C₂²×C₃	semidirect
78. C₂×D₈	C₂²×C₃	semidirect
79. C₂×D₈	C₂²×C₃	semidirect
82. #127	C₂²×C₃	semidirect
83. #124	C₂²×C₃	semidirect
87. #97	C₂²×C₃	semidirect
88. C₄×C₈	C₂²×C₃	semidirect
133. M₁₆	C₃×D₄	semidirect
134. C₂×C₈	C₃×D₄	semidirect
197. C₃	#3155	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×AGL(1,ℤ/8)	2
C₂×C₃×#135	2
C₂²×C₃×D₈	2
C₃×#535	2
C₃×#496	2
C₃×#495	2
C₃×#493	2
C₃×#405	2
#3155	3

Subgroups — C3×#3155 #85019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3155 #85019